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Generalized Boltzmann Equation: Slip-No -Slip

Dynamic Transition in Flows of Strongly

Non-Linear Fluids

Victor Yakhot1, Hudong Chen2, Ilia Staroselsky2,

John Wanderer1 and Raoyang Zhang2,1 Department of Aerospace and Mechanical Engineering,

Boston University, Boston, MA 022152 EXA Corporation, 450 Bedford Street, Lexington, MA 02420

February 2, 2008

Abstract

The Navier-Stokes equations, are understood as the result of the low-order

expansion in powers of dimensionless rate of strain ηij = τ0Sij, where τ0

is the microscopic relaxation time of a close-to- thermodynamic equilibrium

fluid. In strongly sheared non-equilibrium fluids where |ηij | ≥ 1, the hydrody-

namic description breaks down. According to Bogolubov’s conjecture, strongly

non-equlibrium systems are characterized by an hierarchy of relaxation times

corresponding to various stages of the relaxation process. A ”hydro-kinetic”

equation with the relaxation time involving both molecular and hydrodynamic

components proposed in this paper, reflects qualitative aspects of Bogolubov’s

hierarchy. It is shown that, applied to wall flows, this equation leads to qual-

itatively correct results in an extremely wide range of parameter η-variation.

Among other features, it predicts the onset of slip velocity at the wall as an

instability of the corresponding hydrodynamic approximation.

1

1 Introduction

Strongly sheared fluids, in which the usual Newtonian hydrodynamic descriptionbreaks down, are commonly encountered in biology, chemical engineering, micro-machinery design [1]-[2]. Extensive efforts that largely relied upon physical intuitionand qualitative considerations to incorporate corrections at the hydrodynamic levelof description have been made during the years and have achieved various successes.However, these attempts generally fail when the non-linearity of a fluid is strong.Furthermore, theoretical understanding of these highly non-linear systems remains amajor challenge.

The Navier-Stokes equations, which can be derived from the Boltzmann kineticequation as a result of a low -order truncation of an infinite expansion in powers ofdimensionless length-scale Kn , have been extremely successful in describing Newto-nian fluid flows [3]-[7]. The parameter Kn is defined as a ratio between the so calledrelaxation time τ0 associated with molecular collisions and the time of the molecularconvection, namely

Kn = τ0/(L/cs) = τ0cs/L

where L is a characteristic length scale a flow inhomogeneity, and cs is the soundspeed (i.e., an average speed of the molecules). According to kinetic theory, parameterλ = τ0cs represents the so called mean free path in a system and τ0 is the time-scale forthe system to relax to its local thermodynamic equilibrium. Since the ratio L/cs is acharacteristic time-scale of deviation from equilibrium due to density (concentration)perturbations, the Knudsen number Kn is a measure of departure of a fluid systemfrom thermodynamic equilibrium in [4]-[6].

For the purpose of understanding the effects of non-linearity, it is desirable tore-express Kn in terms of the rate of strain (velocity gradient) in the fluid. Using theestimation for the hydrodynamic (macroscopic) length scale,

L ∼ U/|S|

we can also write Kn in an alternative form,

Kn = τ0|S|/Ma ≡ η0/Ma (1)

In the above, U represents the characteristic velocity, and |S| is the magnitude of the

rate of strain tensor Sij of the flow field (|S| =√

2SijSij). Ma = U/cs is the Machnumber. The rate of strain tensor is commonly defined as,

Sij =1

2[∂ui

∂xj

+∂uj

∂xi

]

2

The dimensionless rate of strain, η0 ≡ τ0|S| (in what follows we, instead of η0, willuse the parameter η = τS where τ is a physically relevant relaxation time) canbe viewed as a measure of degree of inhomogeneity and shear in the flow. It isequivalent to a parameter η ∼ W ≡ Uτ0/L commonly used in hydrodynamics ofpolymer solutions. Eqn.(1) indicates that Kn can be large if η is large, speciallyfor low Mach number flows. In many situations η is not a small parameter and theNewtonian fluid based description breaks down. For example, capillary flows or bloodflows through small vessels, rarified gases and granular flows cannot be quantitativelydescribed as Newtonian fluids. A finite η can either be a result of a large relaxationtime associated with the intrinsic fluid property (such as in some polymer solutions),or a result of strong spatial variations like turbulent and micro or nano-flows. Thislatter effect is particularly important at a solid wall where the velocity gradient isoften quite significant. It is known that the no-slip boundary condition is only validin the limit of vanishing Kn. Experimental data indicate that the no-slip conditionbreaks down when Kn ≥ 10−3 [8] and the Navier-Stokes description itself becomesinvalid at Kn ≥ 0.1. For example, the experimentally observed velocity profile in asimple granular Couette flow does not resemble the familiar linear variation of velocitypredicted from the Navier-Stokes equation with no-slip boundary conditions [9]. Thewall slip is an indication of an extremely strong local rate of strain.

The Navier-Stokes equations can be perceived as a momentum conservation lawwith a linear (Newtonian fluid) stress-strain relation [3], [5]-[7], i.e. the deviatoricpart of the stress tensor takes on the following form,

σij ≡ 〈v′

iv′

j〉 −1

3〈v′2〉δij ≈ νSij , i, j = x, y, z (2)

where coefficient ν is the kinematic viscosity. In the above, v′ (≡ v−u) is the relativevelocity between the velocity of molecules (v) and their locally averaged velocity (u)namely the fluid velocity. It is expected that v′ and u represent, respectively, the fast(kinetic) and the slow (hydrodynamic) velocity fields. In an unstrained flow wherethe rate of strain is equal to zero (Sij = 0) , the relation (2) is simply interpreted asthe first term of an expansion in powers of small rate of strain. Therefore, to describerheological or micro-flows with high rate of strains, such a linear approximation mustbe modified to include non-linear effects. However, this task is highly non-trivial ifnot impossible.

The hydrodynamic approximations can be derived from kinetic equations for thedistribution function with the intermolecular interactions accounted for through theso called collision integral [3], [5]-[7]. One can formally write the Bogolubov chainof equations for the distribution function and, in addition to the high powers of thedimensionless rate-of-strain η, generate an infinite number of equations for the multi-particle contributions to collision integral [3],[10]. For a strongly sheared flows where

3

the expansion parameter η is of order unity, this chain cannot be truncated. Anadditional difficulty is that the consistent expansion includes the high-order spatialderivatives , which unfortunately means that, even if we were able to develop theprocedure, the resulting un-truncated infinite- order hydrodynamic equation requiringan infinite set of initial and boundary conditions would be useless.

In this paper, based on Bogolubov’s concept of the hierarchy of relaxation times,we propose a compact representation of the infinite- order hydrodynamic formulationin terms of a simple close-form hybrid (“hydro-kinetic”) equation. The power of theapproach is demonstrated on a classical case of stronly sheared non-Newtonian fluids.It is shown that in our “hydro-kinetic” approach, the formation of a slip velocityat a solid wall and the simultaneous flattening of velocity distribution in a bulk isa consequence of an “instability” of the corresponding hydrodynamic equation withthe no-slip boundary conditions. This “instability” is the result of the non-universaldetails of the flow such as the local values of dimensionless rate of strain η.

2 Basic formulation

Following the standard Boltzmann kinetic theory, we introduce a density distributionfunction f(x,v, t) in the so called single particle phase space (x,v) (≡ Γ). A formallyexact kinetic equation, involving an unspecified collision integral C, can be given [11]

∂tf + vi

∂

∂xi

f = C (3)

Generally speaking, the detailed expression for C involves multi-particle distribution(correlation) functions f (n)(Γ1,Γ2, . . . ,Γn, t) (with n > 1). This results in the famousBogolubov chain of infinite number of coupled equations [3],[10] . This chain cannotbe closed when fluid density is not small. On the other hand, the collision integralC can be modeled in a relatively simple form in the case of a rarified gas where onlythe binary collisions are important [ 11]

C =∫

w′(f ′f ′

1 − f f1) dΓ1dΓ′dΓ′

1 =∫

urel(f′f ′

1 − f f1)dσd3p (4)

In this model, C depends only on the single particle distributions. Here C dΓ is therate of change of the number of molecules in the phase volume dΓ = d3xd3v, urel isthe relative velocity of colliding molecules and dσ and p are the differential scatteringcross-section and momentum of the molecules, respectively. The state variable Γ

describes all degrees of freedom of a molecule and w′ stands for the probability of atransition of two molecules initially in states Γ and Γ1 to states Γ′ and Γ1

′ as a resultof collision. The kinetic equation (3) together with the specific collision integral (4)forms the celebrated Boltzmann equation.

4

It is well known that Boltzmann equation admits an H-theorem in that the systemmonotonically decays (relaxes) to its thermodynamic equilibrium. The correspondinglocal thermodynamic equilibrium distribution function, f eq, is determined from thesolution C(f eq) = 0. If deviation from the equilibrium is small, we can write f =f eq + δf and

C = −f

τ0+

∫

urelf′f ′

1dσd3p ≈ −f − f eq

τ0(5)

with1

τ0

≈∫

urelfeqdσd3p

Equation (5) is often referred to as the “BGK” (mean-field) approximation [12] whichis a natural reflection of the Boltzmann H-theorem. Furthermore, when perturbationfrom equilibrium is weak, it indicates that the relaxation to equilibrium is realized foreach of the distribution functions individually having a common relaxation time. Atthis point, it is important to make the following clarification: Even though as shownabove that (5) was deduced from the Boltzmann binary collision integral model, itsapplicability can be argued to go beyond the low density limit. Indeed, the modelprocess is consistent to the more general principle of the Second law of thermody-namics: A perturbed fluid system always monotonically relax to thermal equilibrium,regardless whether the system has low or high density. Furthermore, in accord withBogolubov [10] the general process of return-to-local equilibrium is true even whenthe deviation from equilibrium is not small.

In the classical kinetic theory, the relaxation time τ0 (≈ const) represents a char-acteristic time of the relaxation process in a weakly non-homogeneous fluid and thesmaller value of τ0, the faster the process of return to equilibrium. The formal expan-sion of kinetic equations in powers of dimensionless rate of strain (Chapman-Enskog(CE) expansion [3], [6]) developed many years ago, leads to hydrodynamic equationsfor the macroscopic (”slow”) variables. When applied to (3), the first order trancationof the formally infinite expansion gives the Navier-Stokes equations with kinematicviscosity:

ν0 = τ0θ. (6)

Development of the expansion based on the full Boltzmann equation is an extremelydifficult task. On the other hand, the simplified Boltzmann-BGK equation (5) for thesingle-particle distribution function allows extension of the Chapman-Enskog expan-sion to include higher powers of dimensionless parameter η. Considering the higher-order terms of the Chapman-Enskog expansion, a simple scalar relaxation term in (5)combined with advection contribution, is expected to generate an infinite number ofanisotropic contributions as contracted products of Sij. Indeed, by expanding equa-tion (5) up to the second order in the Chapman-Enskog series, we can explicitly show

5

that the deviatoric part of the momentum stress tensor takes on the following form[13]

σij = 2τθSij − 2τθ(∂t + u · ∇)(τSij)

−4τ 2θ[SikSkj −1

dδijSklSkl] + 2τ 2θ[SikΩkj + SjkΩki] (7)

where the vorticity tensor is defined as,

Ωij =1

2[∂ui

∂xj

−∂uj

∂xi

]

Note the first O(η) term in (7), resulting from the first order Chapman-Enskog (CE)expansion, corresponds to the Navier-Stokes equations for Newtonian fluid, while thenon-linear corrections to the Navier-Stokes hydrodynamics are generated in the next(O(η2)) order of the CE expansion. It is important to further point out that thememory effects, which appeared in the hydrodynamic approximation (7) as a resultof the second-order CE expansion, are contained in a simple equation (5) which canbe regarded as a generating equation for hydrodynamic models of an arbitrary-ordernon-linearity .

The hydrodynamic approximation (6), (7) has been derived from the equation(5) for the single-particle distribution function valid for a weakly non-equilibriumfluid (small η). In strongly sheared fluids both the assumption of local equilibriumand the low-order trancation of the Bogolubov chain (single-particle collisions) failand the accurate resummation of the series is impossible. Indeed, even in the low-density fluids, the strong shear (η ≥ 1) introduces local fluxes facilitating long-rangecorrelations between particles, invalidating the fluid description in terms of the single-particle distrubution functions. To account for these effects, we have to modify kineticequation (5) in accord with some general ideas about the relaxation processes.Our goal is to reformulate the kinetic equation (5), by modifying the relaxation timeτ and come up with the effective kinetic equation qualitatively accounting for theeffects of the neglected high-order contributions to the Bogolubov chain dominatingthe dynamics far from equilibrium.

In his seminal 1946 book [10] Bogolubov proposed the hierarchy of the time-scalesthat describe relaxation to equilibrium for a system initially far from equilibrium.According to his picture, these initially strong deviations from equilibrium rapidlydecrease, thus allowing an accurate representation of the entire collision integral interms of the single-particle distribution functions. This dramatically simplified repre-sentation is sufficient for an accurate description of the later, much slower, processof relaxation to thermodynamic equilibrium.

6

To make this plausable assumption operational, we have to represent a Bogolubovhierarchy of relaxation times in terms of observable dynamical variables characterizingthe degree of deviation from equilibrium. Since the most natural parameter governingthe dynamics far from equilibrium is the rate of strain |Sij|, we propose that boththe close-to-equilibrium relaxation time τ0 and inverse rate-of-strain 1/|Sij| define theBogolubov hierarchy of the relaxation times. The simplest Galileo invariant relaxationmodel that is compatible with the above physical considerations is:

1

τ=

√

1

τ 20

+ γ|Sne|2 ≡1

τ0

√

1 + γη2ne (8)

where we define in accord with (2), Sneij ≈

σij

ν0

and ηne = τ0|Sne| (i 6= j). In what

follows, wherever it cannot lead to misunderstandings, we will often omitt the suffixne. The new collision integral (i.e., (5) with τ0 replaced by τ in (8)) now describes arelaxation process that involves a rate of strain-dependent relaxation time.

The proposed hydro-kinetic model ((3)-(8)) is chosen to reflect some of the princi-ple elements in the Bogolubov hierarchy. That is, far from equilibrium where the rateof strain is large, the essential time-scale τ is dominated by |S|−1 which corresponds toa rapid first stage of the relaxation process. Later on, when the rate of strain becomessmall, the relaxation process is governed by a close-to-equilibrium relaxation time τ0,as used in the conventional BGK equation (5) leading to the Navier-Stokes formula-tion. It is clear from relation (7) that even though, by restricting our model to thescalar relaxation times in (8), the anisotropic contributions to the stress do appear inthe hydrodynamic description which is a result of the Chapman- Enskog expansion.Since the rate of strain is a property of the flow, the model (8) which includes bothmolecular and hydrodynamic features can be called a hybrid hydro-kinetic approachto non-linear fluids. It is shown below that even with such a specific form in (8), thehydro-kinetic equation is capable of producing some quite non-trivial but physicallysensible results at the hydrodynamic level.

To sumamrise the main points of this section: it is analytically impossible andpractically useless to attempt to describe the strongly non-linear (non-equilibrium)flow physics at the hydrodynamic level. As indicated above, not only the resulting“differential equation” does not have a finite form, it also requires an infinite numberof boundary conditions. The reason for this fundamental difficulty is that the expan-sion in powers of parameter η becomes invalid when η is not small. Thus, to dealwith strongly sheared and time-dependent non-linear flows, it is desirable to use thesimple “hydro-kinetic” description (3)-(8). Clearly, this hybrid representation has afinite form, while it formally contains all the terms in the infinite expansion for thehydrodynamic level. As argued above, the hydro-kinetic formulation is applicable toboth large and small η, corresponding to large and small deviations from equilibrium.

7

3 Wall flows of non-Newtonian fluids

To illustrate the benefits of the hybrid representation of (3)-(8), let us first consider alaminar unidirectional flow in a channel between two plates separated by distance 2Hand driven by a constant gravity force g. In a steady state, the Navier-Stokes equationfor the channel flow can be derived from (5), (8) in the lowest order of the expansionin powers of dimensionless rate of strain η = τ |Sij|. Repeating the procedure leadingto (7) and restricting the expansion by the first term gives the Navier-Stokes equationhaving a ”renormalized” viscosity corresponding to (8) ,

∂tu − g = ∂yν∂yu (9)

whereν = τθ = ν0/

√

1 + γτ 20 |S|

2 (10)

for the special unidirectional situation, |S| = |∂u/∂y|. In the above, we have chosenu to be the streamwise velocity component, while coordinates x and y are along thestreamwise and normal directions of the channel, respectively.

One important point must be mentioned: the expansion in powers of η0 = τ0Scorresponds to the classic CE expansion. The equation (9) has been derived by

expanding in powers of η = τ |S| = η0/√

1 + γη20, which means that even the low-

order perturbation theory in powers of the ”dressed ” parameter η corresponds to aninfinite series in powers of ”bare” parameter η0. Thus, it is extremely interesting toassess the accuracy of the derived hydrodynamic approximation (9)-(10).

Subject to no-slip boundary conditions, the exact steady-state analytic solutionfor (9) is given by:

u(y) =gH2

ν0β(

√

1 − β(y

H)2 −

√

1 − β) (11)

where β ≡ γ(τ0gH/ν0)2 is a dimensionless parameter which can be either positive

or negative, for γ can in principle have either positive or negative signs. One canimmediately see that this particular flow solution breaks down for β > 1. A directindication of this is that the steady state Navier-Stokes equation (9) is no longervalid to describe such a flow, and we have to return back to the full hydro-kineticrepresentation (3)-(8). It is interesting that the equation (9)-(10) allows an unsteadysingular solution:

u = gt + u0φ(y) (12)

where φ(y) = 1 for |y| < H and φ(y) = 0 on the walls where |y| = H . The transitionbetween the two (no-slip (11) and slip (12)) solutions will be demonstrated below.

8

In the rest of the paper, we present solution of the full hydro-kinetic system for thechannel flow for the entire range of parameter variation of γ with initial velocity profileu(y, t = 0) = 0. For this purpose, equations (3)-(8) have been numerically solvedusing a Lattice Boltzmann (LB) algorithm [14]. On each time step the relaxationtime in (8) was calculated with the non-equilibrium rate of strain Sne

ij , defined as:

Sneij =

1

ν0

∫

dvidvj(vi − ui)(vj − uj)(f − f eq) (13)

with i 6= j and u standing for the local value of the mean velocity. It is clear fromthis definition that in thermodynamic equilibrium, the rate-of-strain Seq

ij = 0 and,according to (7), not far fom equilibrium Sij ≈ Sne

ij . The computationally effectiveand widely tested “bounce-back” collision process giving, on the hydrodynamic levelof description, rise to the no-slip boundary condition in the Kn → 0 limit, was imposedon a solid wall. According to the ”bounce back” algorithm, the momentum of the”molecule” colliding with a solid surface changes according to the rule: p ⇒ −p.

When parameter β = 0 , the familiar steady state parabolic solution u(y) =gH2

ν0

(1 − ( y

H

2) was readily derived. Figs.1 present the analytical (i.e., (11)) and thenumerical (i.e., (3)-(8)) solutions of the velocity profiles in the plane channel flowfor, respectively β = 0.4. As we can see, for this value, the difference between thesolutions of (11) and simulations of the full hydro-kinetic model ((3)-(8)) is negligible.This means that in this regime, the lowest order trancation of the Chapman-Enskogexpansion in powers of the ”dressed” parameter η (7) is extremely accurate. Thesame conclusion was shown to hold for all β < 0.5.

The numerical results for β ≥ 0.51 revealed an interesting instability theoreticallypredicted , for β ≥ 1. The results of very accurate numerical simulation (960 cellsacross the channel) are presented on Figs. 2-3.

We can notice qualitatively new features not captured by the steady-state hy-drodynamic approximation: initially, formation of aly narrow wall-boundary layer,accompanied by a strong flattening of the velocity profile in the bulk can be observed.

Later in evolution, the boundary layer becomes unstable with formation of the slipvelocity at the wall. The flow accelerates, eventually becoming a free -falling plugflow, predicted by equation (12). All these phenomena have been experimentallyobserved in the flows of rarified gases and granular materials.

A clarification of the set up is in order: The simulations were performed on an ef-fectively infinitely long (periodic boundary conditions along the streamwise direction)channel, and the flow was driven by an externally imposed constant gravity. This setup differs quite substantially from a pressure-gradient-driven flow of a nonlinear fluidwhere a steady state can be achieved by formation of a non-constant (x-dependent)streamwise pressure gradient. Unlike pressure, gravity is not a dynamical variableand hence the flow lacks the mechanism for establishing a force balance needed to

9

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Y

U

Figure 1: Steady state velocity profile: comparison of hydrodynamic solution Eq. 11(×) with the LBM simulation . β = 0.4.

achieve a steady state. This can be associated with the experimentally observed in-ability of the gravity-driven granular flows in ducts to reach steady velocity profiles[14]. In accord with this theory, the steady velocity profile similar to those shownon Figs. 2-3 can also be observed in a gravity- driven finite-length-pipe or channelflows . In this case we expect the velocity distribution to vary with the length of thepipe/channel.

10

−1 −0.5 0 0.5 10

0.005

0.01

0.015

Y

U

Figure 2: Short-time evolution of unstable velocity profile β = 0.51. The values ofdimensionless time in arbitrary units: from bottom to top: T=0; 2; 4; 6.

4 Conclusion

It has recently been shown that the Lattice Boltmann ( BGK ) equation with theeffective strain-dependent relaxation time can be used for accurate description ofhigh Reynolds number turbulent flows in complex geometries [16]. In this work, thisconcept has been generalized to flows of strongly non-linear fluids. Although thesimple relaxation model (8) was proposed here on a qualitative basis, it has shown tobe capable of producing non-trivial predictions for flows involving strong non-linearity.To the best of our knowledge, this hybrid (”hydro-kinetic”) model is the first attemptof incorporating the principle elements of the Bogolubov conjecture about infinitehierarchy of relaxation times. The most interesting result of application of this modelis the appearance of the slip velocity on the wall as a result of dynamic transitiondriven by increasing rate of strain. Since this transition depends on the wall geometry,it cannot be universal. Thus, to predict this extremely important effect, the model (3)-(8) does not require empirical, externally imposed boundary conditions. The classicalincompressible hydrodynamics relies upon one externally determined parameter, theviscosity coefficient which can be obtained either theoretically (sometimes) or fromexperimental data. The hydro-kinetic approach proposed in this paper needs a singleadditional parameter γ describing physical properties of a strongly non -linear fluid, which can readily be established from a low Reynolds number flow in a capillar by

11

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

Y

U

Figure 3: Long time velocity profile evolution β = 0.51. The values of dimensionlesstime in arbitrary units: from bottom to top: T=0; 8; 16;32;.........80.

comparing the measured velocity profile with the theoretical prediction (11). Furtherapplications of the model (3)-(8) to the separated highly non-linear flows, will showhow far one can reach using this simple approach.

Acknowledgements. One of the authors (VY) has greatly benefitted from stim-ulating discussions with R. Dorfman, I. Goldhirsch, K.R. Sreenivasan, W. Lossert, D.Levermore, A.Polyakov.

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